Wilson renormalization of a reaction–diffusion process

Healthy and sick individuals (A and B particles) diffuse independently with diffusion constants DA and DB. Sick individuals upon encounter infect healthy ones (at rate k), but may also spontaneously recover (at rate 1/τ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρc of the spatially averaged total density. The epidemic evolves as the diffusion–reaction–decay process A+B→2B,B→A, for which we write down the field theory. The stationary-state properties of this theory when DA=DB were obtained by Kree et al. The critical behavior for DA<DB is governed by a new fixed point. We calculate the critical exponents of the stationary state in an ε expansion, carried out by Wilson renormalization, below the critical dimension dc=4. We then go on to obtain the critical initial time behavior at the extinction threshold, both for DA=DB and DA<DB. There is nonuniversal dependence on the initial particle distribution. The case DA>DB remains unsolved.